Model reduction via parametrized locally invariant manifolds: Some examples

Computer Methods in Applied Mechanics and Engineering

Volumn 195, Issue 44-47, 2006, Pages 6287-6311

  Sawant, Aarti ^a   Acharya, Amit ^a  

^a CARNEGIE MELLON UNIVERSITY   (United States)

Author keywords

Invariant manifold; Model reduction; Multiscale modeling; Nonlinear dynamics

Indexed keywords

COMPUTATIONAL METHODS; DYNAMICS; MATHEMATICAL MODELS; NONLINEAR SYSTEMS;

INVARIANT MANIFOLD; MODEL REDUCTION; MULTISCALE MODELING;

ORDINARY DIFFERENTIAL EQUATIONS;

COMPUTATIONAL METHODS; DYNAMICS; MATHEMATICAL MODELS; NONLINEAR SYSTEMS; ORDINARY DIFFERENTIAL EQUATIONS;

EID: 33745946430     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2005.12.012     Document Type: Article

References (31)

1. Acharya A. Parametrized invariant manifolds: a recipe for multiscale modeling. Comput. Methods Appl. Mech. Engrg. 194 (2005) 3067-3089
2. A. Acharya, A. Sawant, On a computational approach for the approximate dynamics of averaged variables in nonlinear ODE systems: toward the derivation of constitutive laws of the rate type, 2005, Preprint.
3. Amirat Y., Hamdache K., and Ziani A. Homogenisation of parametrised families of hyperbolic problems. Proc. Roy. Soc. Edinburgh 120A (1992) 199-221
4. Aubry S., and Le Daeron P.Y. The discrete Frenkel-Kontorova model and its extensions. Physica D 8 (1983) 381-422
5. Aubry S., and Abramovici G. Chaotic trajectories in the Standard Map. The concept of anti-integrability. Physica D 43 (1990) 199-219
6. Aubry S. Anti-integrability in dynamical and variational problems. Physica D 86 (1995) 284-296
7. Bornemann F. Homogenization in time of singularly perturbed mechanical systems. Lecture Notes in Mathematics vol. 1687 (1998), Springer-Verlag, Berlin, Heidelberg, New York
8. Chorin A., Hald O., and Kupferman R. Optimal prediction and the Mori-Zwanzig representation of irreversible processes. Proc. Natl. Acad. Sci. USA 97 (2000) 2968-2973
9. Chorin A., Hald O.H., and Kupferman R. Optimal prediction with memory. Physica D 166 (2002) 239-257
10. Dellnitz M., and Hohmann A. A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer. Math. 75 (1997) 293-317
11. Dieci L., and Lorenz J. Computation of invariant tori by the method of characteristics. SIAM J. Numer. Anal. 32 (1995) 1436-1474
12. Foias C., Sell G.R., and Temam R. Inertial manifolds for nonlinear evolutionary equations. J. Diff. Eqns. 73 (1988) 309-353
13. Foias C., Jolly M.S., Kevrekidis I.G., Sell G.R., and Titi E.S. On the computation of inertial manifolds. Phys. Lett. A 131 (1988) 433-436
14. Givon D., Kupferman R., and Stuart A. Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity 17 (2004) R55-R127
15. Gorban A., and Karlin I.V. Thermodynamic parametrization. Physica A 190 (1992) 393-404
16. Gorban A., Karlin I., and Zinovyev A.Yu. Invariant grids for reaction kinetics. Physica A 333 (2004) 106-154
17. J. Guckenheimer, A. Vladimirsky, A fast method for approximating invariant manifolds, 2003, Preprint.
18. Hughes T.J.R., Calo V.M., and Scovazzi G. Variational and multiscale methods in turbulence. In: Gutkowski W., and Kowalewski T.A. (Eds). Mechanics of the 21st Century. Proceedings of the 21st International Congress of Theoretical and Applied Mechanics (2005), Springer, Warsaw 153-163
19. Jiang B. The Least-Squares Finite Element Method (1998), Springer, New York
20. Jones C.K.R.T. Geometric singular perturbation theory. Dynamical Systems, Montecatini Terme, 1994. Lecture Notes in Mathematics vol. 1609 (1995), Springer-Verlag, Heidelberg, New York 43-118
21. Krauskopf B., and Osinga H.M. The Lorenz manifold as a collection of geodesic level sets. Nonlinearity 17 (2004) C1-C6
22. Lorenz E. Deterministic nonperiodic flow. J. Atmos. Sci. 20 (1963) 130-141
23. Menon G. Gradient systems with wiggly energies and related averaging problems. Arch. Ratl. Mech. Anal. 162 (2002) 193-246
24. J. Moser, Minimal foliations on a torus, in: Topics in Calculus of Variations, Springer Lecture Notes in Mathematics, vol. 1365, 1988, pp. 62-99.
25. Muncaster R.G. Invariant manifolds in mechanics. I. The general construction of coarse theories from fine theories. Arch. Ratl. Mech. Anal. 84 (1983) 353-373
26. Press W.H., Teukolsky S.A., Vetterling W.T., and Flannery B.P. Numerical Recipes in Fortran 77 The Art of Scientific Computing (1999), Cambridge University Press
27. Roberts A.J. Low-dimensional modeling of dynamical systems applied to some dissipative fluid dynamics. In: Ball R., and Akhmediev N. (Eds). Nonlinear Dynamics from Lasers to Butterflies. Lecture Notes in Complex systems vol. 1 (2003), World Scientific 257-313
28. Sacker R.J. A new approach to the perturbation theory of invariant surfaces. Commun. Pure Appl. Math. XVIII (1965) 717-732
29. A. Sawant, Parameterized locally invariant manifolds: a tool for multiscale modeling, Ph.D. thesis, Carnegie Mellon University, 2005.
30. Tartar L. Memory effects and homogenization. Arch. Ratl. Mech. Anal. 111 (1990) 121-133
31. Truesdell C., and Muncaster R.G. Fundamentals of Maxwell's Kinetic Theory of a Simple Monoatomic Gas: Treated as a Branch of Rational Mechanics (1979), Academic Press, New York